August i, 1907] 



NA JURE 



329 



protuborant at llie equator than the figure of the planet 

 on which it rests. The primary effect of the rotation of 

 the earth upon the distribution of continent and ocean 

 is to draw the ocean towards the equator, so as to tend 

 to expose the Arctic and Antarctic regions. We have seen 

 that both Arctic and Antarctic are parts of the continental 

 region. But there is an important secondary effect. 

 l."nder the influence of the rotation the parts of greater 

 density tend to recede further from the axis than the parts 

 of le?;s density. If the density is greater in one hemisphcroid 

 than in the other, so that the position of the centre of 

 gravity is eccentric, the effect must be to produce a sort 

 of furrowed surface ; and the amount of elevation and 

 depression so produced can bo described by an exact mathe- 

 matical formula. It has been proved that this formuhi 

 is a sort of e.xpression which mathematicians name a 

 spherical harmonic of the third degree. 



The shape of the earth is also influenced by another 

 circumstance. We know that at one time the moon was 

 much nearer to the earth than it is now, and that the 

 two bodies once rotated about their common centre of 

 gravity almost as a single rigid system. The month was 

 nearh" as short as the day, and the moon was nearlv 

 fixed in the sky. The earth must then have been drawn 

 out towards the moon, so that its surface was more nearly 

 an ellipsoid with three unequal axes than it is now. The 

 primary effect of the ellipsoidal condition upon the disfri- 

 bution of continent and ocean would be to raise the surface 

 above the ocean near the opposite extremities of the 

 greatest diameter of the equator. But, again, ow'ing to 

 the eccentric position of the centre of gravity, there would 

 be an important secondary effect. The gravitational 

 attraction of an ellipsoid differs from that of a sphere, 

 and It may be represented as the attraction of a sphere 

 together with an additional attraction. If the density 

 was greater in one heml-elllpsoid than in the other, the 

 additional attraction would produce a greater effect in the 

 parts where the density was in excess, and the result, 

 just as in the case of rotation, would be a furrowing of 

 the surface. It has been proved that the formula for this 

 furrowing also is expressed by a spherical harmonic of the 

 third degree. 



We are brought to the theory of spherical harmonics 

 and the spherical harmonic analysis. Spherical harmonics 

 are certain quantities which vary in a regular fashion over 

 the surface of a sphere, becoming positive in some parts 

 and negative in others. I spoke just now of making a 

 model of a nearly spherical surface by removing material 

 from some parts and heaping it up on others. Spherical 

 harmonics specify standard patterns of deformation of 

 spheres. For instance, we might remove material over 

 one hemisphere down to the surface of an equal but not 

 concentric sphere (r/. Fig. 5) and heap up the material 

 over the other hemisphere. We should produce a sphere 

 equal to the original but in a new position. The formula 

 for the thickness of the material removed or added is a 

 spherical harmonic of the first degree. It specifies the 

 simplest standard pattern of deformation. Again, we 

 might remove material from some parts of our model 

 and heap it up on other parts so as to convert the sphere 

 Into an ellipsoid. The formula for the thickness of that 

 which is removed or added is a spherical harmonic of 

 the second degree. Deformation of a sphere Into an 

 ellipsoid is the second standard pattern of deformation. 

 The mathematical method of determining the appropriate 

 series of standard patterns is the theory of spherical har- 

 monics. Its importance arises from the result that any 

 pattern whatever can be reached by first making the 

 deformation according lo the first pattern, then going on 

 to make the deformation according to the second pattern, 

 and so on. If we begin with a pattern, for instance the 

 shape cf the earth, which is not a standard pattern, we 

 can find out how great a deformation of each standard 

 pattern must be made in order to reproduce the prescribed 

 pattern. The method of doing this is the method of 

 spherical harmonic analysis. Except in very simple cases 

 the application of it involves rather tedious computations. 

 With much kind assistance and encouragement from Prof. 

 Turner. I made a rough spherical harmonic analysis of 

 the eanh's surface. I divided the surface into 2592 small 

 areas, rather smaller on the average than lireat Britain. 



gave them the value -l-i, or one unit of elevation, if they 

 are above the sea, and the value — i, or one unit of 

 depression, if they are below the 1400-fathom line. To 

 the Intermediate areas I gave the value o. The distribu- 

 tion of the numbers over the surface was analysed for 

 spherical harmonics of the first, second, and thira degrees. 

 Any spherical harmonic of the first degree gives us a 

 division of the surface into two hemispheres — one elevated, 

 the other depressed. The spherical harmonic analysis 

 informs us as to the position of the great circle which 

 separates the two hemispheres, and also as to the ratio 

 of the maximum elevation of this pattern to the maximum 

 elevation of any other pattern. The central region of 

 greatest elevation of this pattern is found to be in the 

 neighbourhood of the Crimea, and the region of elevation 



Fig.6. 



contains the Arctic Ocean and the northern and central 

 parts of the Atlantic, Europe, Africa, Asia, most of North 

 .\merica, and a small part of South America. When the 

 surface is mapped on a rectangle in the same way as 

 before, the chart of the harmonic is that shown in Fig 6.' 

 The actual disproportion between the amounts of con- 

 tinental area in the northern and southern hemispheres is 

 associated with the . result that the central region of 

 elevation, as given by the analysis, is about 45° north of 

 the equator ; and the extension of the Pacific Ocean and 

 adjoining Southern Ocean to much higher southern than 

 northern latitudes is associated with the corresponding 

 position of the central region of greatest depression about 

 45° south of the equator. In regard to harmonics of the 

 second degree, the spherical harmonic analysis informs us 

 as to the ellipticity of the equator and the obliquity of 



Fig. 7. 



the principal planes of that ellipsoid w'hich most nearly 

 represents the elevation of the surface above or its depres- 

 sion below the surface of the ocean, or the geoid. The 

 result is an equatorial region of depression, which spreads 

 north and south unequally in different parts and forms a 

 sort of immense Mediterranean, containing two great 

 basins, and separating a northern region of elevation from 

 a southern. The northern region of elevation occupies the 

 northern part of the Atlantic Ocean and runs down to 

 and across the equator in the neighbourhood of Borneo. 

 The southern region of elevation occupies the southern 

 part of the Pacific Ocean, and it runs up to and across 

 the equator in the neighbourhood of Peru. The chart 

 of the harmonic is shown In Fig. 7. The equatorial regions 



1 In this figure, and in the following figures, regions of elevalun are 

 •haded, and regions of depression are left blank. 



NO. 1970, VOL. 76] 



